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ELSEVIER Computers and Mathematms with Apphcatmns 49 (2005) 1867-1888
www.elsevier com/locate/camwa
Drawing Graphs by Eigenvectors:
Theory and Practice*
Y . KOREN
AT&T Labs--Research
Florham Park, NJ 07932, U S A.
y e h u d a © r e s e a r c h , azZ.
com
(Recewed July 2003, rewsed and accepted August 2004)
Abstract--The
spectral approach for graph vlsuahzatlon computes the layout of a graph using
certain mgenvectors of related matmces Two important advantages of thin approach are an ablhty
to compute optimal layouts (according to specific reqmrements) and a very rapid computation time.
In thin paper, we explore spectral vmuahzatmn techniques and study their properties from different
points of wew We also suggest a novel algorithm for calculating spectral layouts resultmg m an
extremely fast computatmn by optlmmmg the layout within a small vector space (~) 2005 Elsevmr
Ltd All rights reserved
Keywords--Graph
tral graph theory
drawing, Laplaclan, Eigenvectors, Fledler vector, Force-directed layout, Spec-
1. I N T R O D U C T I O N
A graph G(V, E) is an abstract structure that is used to model a relation E over a set V of entities. Graph drawing is a standard means for visualizing relational information, and its ultimate
usefulness depends on the readability of the resulting layout, that is, the drawing algorithm's
ability to convey the meaning of the diagram quickly and clearly. To date, many approaches
to graph drawing have been developed [2,3]. There are many kinds of graph-drawing problems,
such as drawing di-graphs, drawing planar graphs, and others. Here, we investigate the problem of drawing undirected graphs with straight-line edges. In fact, the methods that we utilize
are not limited to traditional graph drawing and are intended, also, for general low dimensional
visualization of a set of objects according to their pairwise similarities (see, e.g., Figure 1).
We have focused on spectral graph drawing methods, which construct the layout using eigenvectors of certain matrices associated with the graph To get some feeling, we provide results
for three graphs in Figure 2. This spectral approach is quite old, originating with the work of
Hall [4] in 1970. However, since then, it has not been used much. In fact, spectral graph drawing
algorithms are almost absent in graph-drawing literature (e.g., they are not mentioned in the
two books [2,3] that deal with graph drawing). It seems that, in most visuahzation research, the
spectral approach is difficult to grasp, in terms of aesthetics. Moreover, the numerical algorithms
for computing the eigenvectors do not possess an intuitive aesthetic interpretation.
*An early and short version of this work appeared m [1]
0898-1221/05/$ - see front matter (~) 2005 Elsevmr Ltd All rights reserved
dol.10 1016/] camwa.2004 08 015
Typeset by .AMe-TEX
1868
Y KOREN
1
0.8
°if
0.6
06
04
0.4
0.2
!
0
Q
,
,
,
0
0.2 0.4 0,6 0.8
1
(a) A drawing usmg the eigenvectors of the Laplaclan
-0.4
~3D
I
.
.
-0.5
.
.
.
~-~
0
..........
0.5
1
(b) A drawing using the degree-normalized eigenvectors
Figure 1. Visualization of 300 odor patterns as measured by an electronic nose
(a) The 4970 graph IVI = 4,970, IEI = 7, 400.
(c) The crack graph [10]. ]V] = 10,240, [E I = 30, 380.
(b) The 4elt graph [10]
45,878.
iV I = 15,606,
IEI
=
(d) A 100 x 100 folded grid with central horizontal
edge removed. IV] = 10,000, ]El = 18,713.
Figure 2 Drawings obtained from the Laplaclan elgenvectors.
Drawing Graphs by Eigenvectors
1869
We believe that the spectral approach has two distract advantages that make it very attractive.
First, it provides us with a mathematically-sound formulation leading into an exact solution to
the layout problem, whereas, almost all other formulations result in an NP-hard problem, which
can only be approximated. The second advantage is computation speed. Spectral drawings can
be computed extremely fast as we show in Section 7. This is very important because the amount
of information to be visualized is constantly growing rapidly.
Spectral methods have become standard techniques in algebraic graph theory (see, e.g., [5]).
The most widely used techniques utilize eigenvalues and eigenvectors of the adjacency matrix of
the graph. More recently, the interest has shifted somewhat to the spectrum of the closely related
Laplacian. In fact, Mohar [6] claims that the Laplacian spectrum is more fundamental than this
of the adjacency matrix
Related areas where the spectral approach has been popularized include clustering [7], partitioning [8], and ordering [9]. However, these areas use discrete quantizations of the eigenvectors,
unlike graph drawing, which employs the eigenvectors without any modification. Regarding this
aspect, it is more fundamental to explore properties of graph-related eigenvectors in the framework of graph drawing.
In this paper, we explore the properties of spectral visualization techniques, and provide different explanations for their ability to draw graphs nicely. Moreover, we consider a modified
approach that uses what we will call degree-normalzzed ezgenvectors, whmh have aesthetic advantages in certain cases. We provide an aesthetically-motivated algorithm for computing the
degree-normalized eigenvectors. We also introduce a novel algorithm for computing spectral layouts that facilitates a significant reduction of running time by optimizing the layout within a
small vector space.
2. B A S I C
NOTIONS
A graph is usually written G(V, E), where V = {1... n) is the set of n nodes, and E is the
set of edges (we assume no self-]oops or parallel edges). Each edge (i,j I is associated with a
nonnegative weight w~3 that reflects the similarity of nodes z and j. Thus, more similar nodes
are connected with "heavier" edges. For unweighted graphs, one usually takes uniform weights.
Let us denote the neighborhood of z by,
N(z) = {3 I (z,3) • E}.
The degree of node z is deg(~) de___fE3EN(~ ) W~:. Throughout the paper, we have assumed, without loss of generality, that G is connected, otherwise, the problem we deal with can be solved
independently for each connected component.
A p-dimensional layout of the graph is defined by p vectors,
xl,
. . . ~X p E ~n
where
x 1 ( ~ ) , . . . , x p (~)
are the coordinates of node ~. In most applications, p ~ 3, but here, we will not specify p, so as
to keep the theory general (however, always p < n). Let us denote by d~3, the Euclidean distance
between nodes ~ and j (in the p-D layout), so
=
-
(3))
k=l
The
ad3acency-matrix of the graph G is the symmetric n x n matrix A G, where
0,
~tE,
AG = {
~,j = 1, . . , n .
w . , (~,3) E E,
We will often omit the G in A e.
1870
Y KOREN
The Laplac~an is another symmetric n x n matrix associated with the graph, denoted by L G,
where
deg(i), ~--2,
L~ =
O,
~ ¢ 3,
(~,3) q~E,
~,3 = 1 , . . . , n .
Again, we will often omit the G in L G.
The Laplacian has many interesting properties (see, e.g., [6]). Here, we state some useful
features.
(1) L is a real symmetric and hence, its n eigenvalues are real and its eigenvectors are orthogonal.
(2) L is positive semidefinite and hence, all eigenvalues of L are nonnegative.
(3) In ~ f (1, 1 , . . . , 1) T E R ~ is an eigenvector of L, with the associated eigenvalue 0.
(4) The multiplicity of the zero eigenvalue is equal to the number of connected components
of G. In particular, if G is connected, then, In is the only eigenvector associated with
eigenvalue 0
The usefulness of the Laplacian stems from the fact that the quadratic form associated with
it is just a weighted sum of all pairwise squared edge-lengths.
LEMMA 1. Let L be an n x n Laplaclan, and let x E •n.
xTLx=
E
Then,
w,(x(i)--x(2)) 2 .
More generally, for p vectors x l , . . . , x p E ~n, we/]ave
P
(xk)TLx k=
k=l
(~,j)EE
The proof of this lemma is direct.
Throughout the paper, we will use the convention, 0 = )u < A2 _< -.. _< A~, for the eigenvalues
of L, and denote the corresponding real orthonormal eigenvectors by
vl=(--~).l~,v2,..
,v ~.
Let us define the degrees matmx as the n × n diagonal matrix D that satisfies D~ -- deg(i).
Given a degrees matrix, D, and a Laplacian, L, then, a vector u and a scalar # are termed,
generalized eigenpairs of (L, D), if Lu = #Du. Our convention is to denote the generalized
eigenvectors of (L,D) by u 1 = a . in,U2,... ,u n, with corresponding generalized eigenvalues
0 = #1 < #2 ~< "'" ~< #n. (Thus, Lu k = Iz~Duk, k = 1 , . . . , n . ) To uniquely define u l , u 2 , . . . , u ~,
we require them to be D-normalized, so,
( u k ) T D u k .~_ l,
k ~- i , . . . n.
We term these generalized eigenvectors, the degree normalized eigenvectors.
In general, for a symmetric (positive-semidefinite) matrix A and a positive-definite matrix B,
it can be shown that all the generalized eigenvalues of (A, B) are real (nonnegative), and that all
the generalized eigenvectors are B-orthogonal. Consequently, (uk)TDu I = 0, V k ~ 1.
2.1. M a t h e m a t i c a l P r e l i m i n a r i e s
Now, we develop some essential mathematical background that is needed for subsequent derivations. Different parts of this material can be found in standard linear algebra textbooks. The
casual reader can make do with understanding the theorems and corollaries, and does not have
to delve into the proofs. In the following, 6~j is the Kronecker delta (defined as 1 for z -- 3 and
as 0, otherwise), and A k is the k th column of matrix A.
Drawing Graphs by Eigenvectors
1871
Given a s y m m e t r i c m a t r i x An×n, denote by v i, . . . , v ~ its eigenvectors, with corresponding eigenvalues Ai ~< " ' ~ An. Then, v i , . . . , v; are an optimal solution o f the constrained
m i n i m i z a t m n problem,
THEOREM
1.
P
min
~
(xk) S
Axk,
(i)
k=l
subject
to:
(xk) T
k,l = 1,...p.
X l = 6kl ,
Throughout this section, we assume p < n. Before providing the proof of Theorem 1, we prove
the following lemma.
LEMMA 2. L e t
v l , . . . , V p - 1 C ~ n be some vectors, and let X be an n x p m a t r i x with orthogonM
columns (i.e., x T x = I ) , then, there is an n x p m a t r i x with orthogonM columns Y that satisfies
the following.
(1) For every 2 <~ k <. p: y k is orthogonal to v l , . . . , v k - i .
(2) For every n x n m a t r i x A, trace ( X T A X ) = trace ( y T A y ) .
PROOF. Let us denote the projection of v 1 into TCange ( X ) (i.e., span ( X 1 , . . , X P ) ) by 01.
If 0 i = 0, then, we set y 1 = X 1 and ]?2,
,])P = X 2, . , X p Certainly ~ 2 , . . . , ~ p are
orthogonal to v 1. Otherwise, (when 0 i ¢ 0), we rotate X 1, X 2 . . . , X p within T~ange ( X ) obtaining yi,IY2 ...,]Yp, such that y i = 0i/[[911[" Since rotations do not alter orthogonality
relations, we still have t h a t ]Y2 . . . , ]?p are orthogonal to v i. We continue recursively with the
vectors v 2 , . . . , v p - i and the matrix (]Y2, . ,]YP). Note that the recursion performs rotations
only within span(Y 2, .., ]YP), so all resulting vectors are orthogonal to v I and to y i . At the
end of the process, we obtain p orthogonal vectors, y i . . . . YP, that satisfy the first requirement
from Y in the lemma.
Since all rotations are performed within Tiange ( X ) , there is some p x p matrix R, such that
Y = X R . Moreover, since X and Y have orthogonal columns, we obtain
I = yTy
= RTxTxR
= RTR.
Hence, R is an orthogonal matrix implying that also R R T = I. We use the fact that the trace is
cyclically-commutative to obtain the following,
trace ( y T A y ) = trace (R T X T A X R ) = trace ( R R T X T A X ) = trace ( Z T A X ) .
I
Now we can prove Theorem 1.
PROOF. Let x i , . . . , x p be arranged column-wise in the n x p matrix X. Now, we can rewrite (1)
in a simpler notation,
min
trace ( X T A X ) ,
x
(2)
subject to.
x T x = I.
Let V0 = (v~,... ,v~) be the minimizer of (2). Since the eigenvectors v i , . . . , v ~ form a basis
of ]Rn, we can decompose every v0k as a linear combination, where v0k -- vA-~l=
' " 1 akvt
l
" Lemma 2
allows us to assume, without loss of generality, that, for every 2 ~< k <~ p: v0k is orthogonal to the
eigenvectors v l, .., v k - i . We may, therefore, take a~ = 0, for l < k, and write,
=
l=k
Next, we use the constraint, (v0k)Tv~ ----1, to obtain an equation for the coefficients a~,
i = (vo
=
(5
\l=k
/ 5
~--
/
\l=k
/
k 2
l=k
where the last equation stems from the orthonormaiity of v i, v 2 , . . , v n.
,
1872
Y KOREN
Hence, ~-~Lk(a~) 2 = I (a generalization of Pythagoras' law). Using this result, we can expand
the quadratic form, (xk)-rAx k,
A
:
\l=k
/
:
\l=k
a vI
/
\l=k
=
C~
\l=k
a Av 1
/
\l=k
V1
/
/
Oz ~ l V l
\l=k
(3)
/
l=k
l=k
---- Ak.
Thus, the target function satisfies
P
trace ( X T A X ) : E
P
(xk) T Ax k >1 E
k=l
Since y~f=I(vk)TAv k = ~'~=1
Ak, we deduce
Ak.
k=l
that the low eigenvectors v l , . . . , v p solve (1).
|
Now, we state a more general form of Theorem 1.
THEOREM 2. Given a symmetric matrix
and a positive definite matrix B, denote by
v l , . . . , Vn the generalized eigenvectors of (A, B), with corresponding eigenvalues A1 ~ " " ~ An.
Then, v l , . . . , v p are an optimal solution of the constrained minimization problem,
Anxn
P
rain
X I , . ,X p
Z (xk)T Axk,
(4)
k=l
subject to:
(xk) T B x l = 6kl,
k,l=l,...p.
PROOF. Since B is positive definite, it can be decomposed into B = c T c , where C is an n x n
invertible matrix. Let us substitute in (4), x k = C - l y k, so now, we reformulate the problem as
P
min
E
(yk) T C_ T A C _ l y k
~l 1 ~ . . ~ y P
k=l
subject to:
( y k ) r y l = 6kl,
(5)
k,l = 1 , . . . p .
Let y0~,..., y~ be the minimizer of (5). By Theorem 1, these are simply the p lowest eigenvectors
of C - T A G -1, therefore, obeying C - T A C - l y ko = Akyko. Using this equation and transforming
back into Xko = C - l y e , we get
c - T Axko = AkCx ko.
This implies Ax~ = AkBx~, so the solvers of (4) are nothing but the lowest generalized eigenvectors of (A, B).
|
Frequently, the following version of Theorem 2 will be the most suitable.
Drawing Graphs by Eigenvectors
1873
COROLLARY 1. Given a s y m m e t r i c matrix A~xn and a positive definite matrix B n x n , denote by
v l , . . . , v ~, the generalized eigenvectors of (A, B), with corresponding eigenvalues A1 <.... <~ AN.
Then, vl, . . . , v p are an optimal solution of the constrained minimization problem,
P
E (xk)~ Axk
k=l
min
p
,
(6)
E (zk)T B~k
k=l
(xl) T Bx 1 =
subject to:
(x2) T B x 2 . . . . .
(xk) T B x l = O ,
(xP) T S z p,
1 <k¢l<p.
(7)
(8)
PROOF. Note that the value of (6) is mvanant under sealing since, for any constant c ¢ O,
k=l
k=l
(~)~ B ~
~ ( ~ ) ~ B (ex~)
k=l
k=l
Thus, we can always scale the optimal solution, so that
(~,)T B~X = (~2)T B ~ . . . . .
(~P)~ B ~ = 1.
This reduces the problem to the form of (4).
It is straightforward to prove the following corollary that deals with a case in which the solution
must be B-orthogonal to the lowest generalized eigenvectors. In this case, we just take the next
low generalized eigenvectors.
COROLLARY 2. Gwen a symmetric matrix Anxn and a positive definite m a t r i x Bnxn~ denote by
vl,..., Vn, the generalized eigenvectors of (A, B ), with corresponding elgenvalues A1 <<.... <~ AN.
Then, v k + l , . . . , v k+p are an optlmM solution of the constrained minimization problem,
P
E (x') T ~ "
min
l=l
p
=1, ,xP
E
subject to:
(x 1)T B x I = (x2) T B x 2 . . . . .
(X') T B x '
(xV)T BxP,
(~')T B~: = 0,
1<<.~¢3<<.p,
(x~)-C B v 3 = 0 ,
~=l,...,p,
3. S P E C T R A L
GRAPH
(9)
3=1,...,k.
DRAWING
The earliest spectral graph-drawing algorithm was that of Hall [4]; it uses the low eigenvectors
of the Laplacian Henceforth, we will refer to this method as the e~genpro3eetwn method. Later, a
similar idea was suggested in [11], where the results are shown to satisfy several desired aesthetic
properties. A few other researchers utilize the top elgenvectors of the adjacency matrix instead of
those of the Laplacian. For example, consider the work of [12], which uses the adjacency matrix
eigenvectors to draw molecular graphs. Recently, eigenvectors of a modified Laplacian were used
in [13] for the visualization of bibliographic networks.
In fact, for regular graphs of uniform degree deg, the eigenvectors of the Laplacian equal
those of the adjacency matrix, but in a reversed order, because L = deg .I - A, and adding
the identity matrix does not change eigenvectors. However, for nonregular graphs, use of the
Laplacian is based on a more solid theoretical basis, and in practice, also gives nicer results than
those obtained by the adjacency matrix. Hence, we will focus on visualization using eigenvectors
of the Laplacian.
1874
Y. KO~EN
3.1. E n e r g y - B a s e d D e r i v a t i o n of t h e E i g e n p r o j e c t i o n M e t h o d
One of the most popular approaches to graph-drawing is the force-directed strategy [2,3] that
defines an energy function (or a force model), whose minimization determines the optimal drawing. Consequently, we will introduce the eigenprojection method as the solution of the following
constrained minimization problem,
E ~,J4
E (x 1, .. x p) de_f(*,j)~E
min
~1 .... ~.
"
subject to:
'
Y~d,~
Vat (x 1) = Var (x 2) . . . . .
Coy (xk, x~) = 0,
(10)
'
Var (xP),
1 < k # z < p.
(11)
(12)
Here, Vat(x) is the variance of x, defined as usual by,
n
Var(x) = 1 E
(x (i) - ~)2
n
where ~, is the mean of x. Also, Cov(x k, x l) is the covariance of x k and x I defined as
1 ~ (~ (~)_~) (x~(i)- ~).
n
z=l
Recall that
p
4 = ~ (x~ ( 0 - ~ (3)) 2
k=l
The energy to be minimized, E ( x l , . . . ,xP), strives to make edge lengths short (to minimize
the numerator) while scattering the nodes in the drawing area preventing an overcrowding of the
nodes (to maximize the denominator). This way, we adopt a common strategy to graph drawing
stating that adjacent nodes should be drawn closely, while, generally, nodes should not be drawn
too close to each other; see, e.g., [14,15]. Since the sum is weighted by edge-weights, "heavy"
edges have a stronger impact and hence, will be typically shorter. The first constraint (11) forces
the nodes to be equally scattered along each of the axes. In this sense, the drawing has a perfectly
balanced aspect ratio The second constraint (12) ensures that there is no correlation between
the axes, so that each additional dimension will provide us with as much new information as
possible 1.
The energy and the constraints are invariant under translation. We eliminate this degree of
freedom by requiring that, for each 1 ~< k ~<p, the mean of x k is 0, i.e.,
n
Ex~(~) = (xk) T . 1 , = 0 .
~=1
This is very convenient since now the no-correlation constraint, Cov (x k, x t) = 0, is equivalent to
requiring the vectors to be pairwise orthogonal (xk)-Cx I = 0. Also, now Var (x k) = 1/n(xk)Tx k,
so the uniform variance constraint can be written in a simple form,
1The strategy to reqmre no correlation between the axes is used in other visualization techniques hke principal
components analysis and classical multidimensional scaling [16].
Drawing Graphs by Exgenvectors
1875
Using L e m m a 1, we can write ~(,,o)eE w~ad~3 in a matrix form,
P
Z (x~)~ ~ .
k=l
Now, the desired layout can be described as the solution of the following equivalent minimization
problem,
P
T
E (~) L~~
mln
x~, , ~
2
~ d, 3
*<a
subjectto:
,
(13)
(x~)~(~)=~,,
k,l=l,...,p,
( ~ k ) r . 1~ = 0,
k = 1 , . . . ,p.
We can simplify the d e n o m m a t o r ~*<a d*2, using the following lemma.
LEMMA 3. Let x E ]Rn, such that X T l n -~ O, then,
n
*=1
~<3
PROOF.
E
(x (,) - x (3)) 5 = ~1
z<,
(x (z) - x (3)) 2 = 21 \ 2n ~=1 x ( ~ ) 2 - 2 , , , < x ( ~ ) x ( j )
%3=1
----n .
x(z) 2 *=1
x (z)
~=1
x ( j ) ----n .
3=1
x (i) 2.
*=1
n
The last step stems from the fact that x is centered, so that Y~3=1 x(3) = 0.
|
As a consequence, we get
P
P
P
Z 4 = Z Z (~ (z)- ~ o)) ~ = Z Z (~ (~)- x~ (~))~= Z ~-(x~)~.
*<3
*<3 k = l
k = l *<3
k=l
Therefore, we rewrite again the minimization problem in an equivalent form,
P
E
subject
(xk) T Lx k
k=l
z~ (xk) T xk '
k=l
min
xl . . . .
to:
(Xk) T Xz = 5kZ,
( x k ) r • 1~ = o,
(14)
k, l = 1 , . . . ,p,
k = 1,... ,p.
Let us substitute A = L, B = I in Corollary 2. Using the fact t h a t the lowest eigenvector
of L is ln, we obtain that the optimal layout is given by the lowest positive Laplacian eigenvectots v 2 , . . . , v p+I The resulting value of the energy is
p+l
k=2
the sum of the corresponding eigenvalues.
Note that an interesting property of the eigenprojection is that the first p - 1 axes of the
optimal p-dimensional layout are nothing but the optimal (p - 1)-dimensional layout.
1876
4.
Y KO~EN
DRAWING
USING
DEGREE-NORMALIZED
EIGENVECTORS
In this section, we introduce a new, related spectral graph drawing method that associates the
coordinates with some generalized eigenvectors of the Laplacian.
Suppose that we weight nodes by their degrees, so the mass of node i is its degree--deg(~).
Now, if we take the original constrained minimization problem (14) and weight sums according to
node masses, we get the following degree-weighted constrained minimization problem (where D
is the degrees matrix),
p
E (z k) T Lxk
mm
~ ....
subject to:
k=l
~ (zk) T Dx~'
(15)
(xk) 7- D (x l) = 5,j,
k,l = 1 , . . . , p ,
(xk) T DI,~ = 0,
k = 1,...,p.
Substitute A = L, B = D in Corollary 2 to obtain the optimal solution u2,... ,u v+l, the generalized eigenvectors of (L, D). We will show that in several aspects using these degree-normalized
eigenvectors is more natural than using the eigenvectors of the Laplacian. In fact, Shi and Malik [7] have already shown that the degree-normalized eigenvectors are more suitable for the
problem of image segmentation. For the visualization task, the motivation and explanation are
very different.
In problem (15), the denominator moderates the behavior of the numerator. The numerator
strives to place those nodes with high degrees at the center of the drawing, so that they are in
proximity to the other nodes. On the other hand, the denominator also emphasizes those nodes
with high degrees, but in the reversed way; it strives to enlarge their scatter. The combination
of these two opposing goals, helps in making the drawing more balanced, preventing a situation
in which nodes with lower degrees are overly separated from the rest nodes.
Another observation is that degree-normalized eigenvectors unify the two common spectral
techniques: the approach that uses the Laplacian and the approach that uses the adjacency
matrix.
CLAIM The generalized eigenvectors of (L, D) are also the generalized eigenvectors of (A, D),
with a reversed order.
PROOF. Utihze the fact that L = D - A. Take u, a generalized eigenvector of (L, D). The
vector u satisfies (D - A ) u = # D u , or equivalently, by changing sides, A u = D u - # D u . This
implies that
A u = (1 - #) D u ,
and the claim follows. The proof in the other direction is performed similarly. Thus, A and L
have the same D-normalized eigenvectors, although the order of eigenvalues is reversed.
|
In this way, when drawing with degree normalized eigenvectors, we can take either the low
generahzed eigenvectors of the Laplacian, or the top generalized eigenvectors of the adjacency
matrix, without affecting the result. (Remark: In this paper, when referring to "top" or "low"
eigenvectors, we often neglect the topmost (or lowest) degenerate eigenvector a . 1~.)
The degree-normalized eigenvectors are also the (nongeneralized) eigenvectors of matrix D - 1 A .
This can be obtained by left-multiplying the generalized eigenequation, A x = # D x , by D -1,
obtaining the eigenequation,
D - l A x = #x.
(16)
Note that D - 1 A is known as the trans~twn matrix of a random walk on the graph G. Hence, the
degree-normalized eigenprojection uses the top eigenvectors of the transition matrix to draw the
graph.
Drawing Graphs by Eigenvectors
1877
Regarding drawing quality, for most unweighted graphs with which we experimented (that are
probably close to being regular), we have observed not much difference between drawing using
eigenvectors and drawing using degree-normalized eigenvectors. However, when there are marked
deviations in node degrees (as is often the case when working with weighted graphs), the results
are quite different. This can be directly seen by posing the problem as in (15). Here, we provide
an alternative explanation based on (16). Consider the two edges el and e2. Edge el is of
weight 1, connecting two nodes, each of which is of degree 10. Edge e 2 is of weight 10, connecting
two nodes, each of whmh is of degree 100. In the Laplacian matrix, the entries corresponding
to e2 are 10 times larger than those corresponding to el. Hence, we expect the drawing obtained
by the eigenvectors of the Laplacian, to make the edge e2 much shorter than el (here, we do not
consider the effect of other nodes that may change the lengths of both edges). However, for the
transition matrix in (16), the entries corresponding to these two edges are the same, hence, we
treat them similarly and expect to get the same length for both edges. This reflects the fact that
the relatwe importance of these two edges is the same, i.e., 1/10.
In many kinds of graphs numerous scales are embedded, which indicates the existence of dense
clusters and sparse clusters. In a traditional eigenprojection drawing, dense clusters axe drawn
extremely densely, while the whole area of the drawing is used to represent sparse clusters or
outliers. This might be the best way to minimize the weighted sum of square edge lengths, while
scattering the nodes as demanded. A better drawing would allocate each cluster an adequate
area. Frequently, this is the case with the degree normalized eigenvectors that adjust the edge
weights in order to reflect their relative importance in the related local scale.
For example, consider Figure 1, where we visualize 300 odors as measured by an electronic
nose. Computation of the similarities between the odors is given in [17]. The odors are known to
be classified into 30 groups, which determine the grayscale of each odor m the figure. Figure la
shows the visualization of the odors by the eigenvectors of the Laplacian. As can be seen, each
of the axes shows one outlying odor, and places all the other odors about at the same location.
However, the odors are nicely visualized using the degree normalized eigenvectors, as shown in
Figure lb.
5. A D I R E C T C H A R A C T E R I Z A T I O N
OF S P E C T R A L
LAYOUTS
So far, we have derived spectral methods as solutions of optimization problems. In this section,
we characterize the eigenvectors themselves, in a rather direct manner, to clamfy the aesthetic
properties of the spectral layout. The new derivation will show a very simple relation between
layout aesthetics and the Laplacian (generalized) eigenvectors. Here, the degree-normalized eigenvectors will appear as a more natural way for spectral graph drawing.
The quadratic form associated with the Laplaclan,
ZLx
:
(x
Z
-
m tightly related to the aesthetic criterion that calls for placing each node at the weighted centroid
of its neighbors. To conceive this note that, for each node i differentiating xTLx with respect
to x(i) gives
OxTLx
o x (i)
-
2
w,j (x
- x (3)).
Equating this to zero and isolating x ( J , we get
w,~x (3)
x (~) = jcx(~)
deg (~)
1878
Y KOREN
Hence, when Mlowing only node ~ to move, the location of ~ that minimizes x T L x is the weighted
centroid of z neighbors.
When the graph is connected, placing each node at the weighted centroid of its neighbors can be
strictly achieved only by the degenerate solution that puts all nodes at the same location. Hence,
to incorporate this aesthetic criterion into a graph drawing algorithm, it should be modified
appropriately.
Presumably, the earliest graph drawing algorithm, formulated by Tutte [18], is based on placing
each node on the weighted centroid of its neighbors. To avoid the degenerate solution, Tutte
arbitrarily chose a certain number of nodes to be anchors, i.e., he fixed their coordinates in
advance. Those nodes are typically drawn on the boundary. This, of course, prevents the collapse;
however, it raises new problems, such as which nodes should be the anchors, how to determine
their coordinates, and why, after all, such an anchoring mechanism should generate nice drawings.
An advantage of Tutte's method is that in certain cases, it can guarantee achieving a planar
drawing (i.e., without edge crossings).
Thtte treats in different ways the anchored nodes and the remaining nodes. Whereas, the
remaining nodes are located exactly at the centrold of their neighbors, nothing can be said about
anchored nodes. In fact, in several experiments, we have seen that the anchored nodes are located
quite badly.
Alternatively, we do not use different strategies for dealing with two kinds of nodes, but rather,
we treat all the nodes similarly. The idea is to gradually increase the deviations from centroids
of neighbors as we move away from the origin (that is the center of the drawing). This reflects
the fact that central nodes can be placed exactly at their neighbors' centroid, whereas boundary
nodes must be shifted outwards.
More specifically, node i, which is located in place x(i), is shifted from the center toward the
boundary by the amount of #. Ix(i)l, for some # > 0. Formally, we request the layout x to satisfy,
for every 1 ~< i ~< n,
E
(3)
X (~) -- 9eY(~)
deg
= '"
Note that the deviation from the centroid is always toward the boundary, i.e., toward +c~ for
positive x(i) and toward - o o for negative x(z). In this way, we prevent a collapse at the origin.
We can represent all these n requests compactly in a matrix form, by writing,
D - 1 L x = #x.
Left-multiplying both sides by D, we obtain the familiar generalized eigenequation,
Lx = ~Dx.
We conclude with the following important property of degree-normalized eigenvectors.
PROPOSITION 1. Let u be a generahzed elgenvector of (L, D), with associated eigenvalue #.
Then, for each i, the exact deviation from the centroid of neighbors is
deg (i)
= #" u (i).
Note that the eigenvalue # is a scale-independent measure of the amount of deviation from
the centroids. This provides us with a fresh new interpretation of the eigenvalues that is very
different from the one given in Section 3.1, where the eigenvalues were shown as the amount of
energy in the drawing.
Thus, we deduce that the second smallest degree-normalized eigenvector produces the nondegenerate drawing with the smallest deviations from centroids, and that the third smallest
degree-normalized eigenvector is the next best one and so on.
Similarly, we can obtain a related result for eigenvectors of the Laplacian.
Drawing Graphs by Elgenvectors
1879
PROPOSITION 2. Let v be an eigenvector of L, with associated eigenvalue A. Then, for each z,
the exact deviation from the centroid of neighbors is
w~v (3)
v(~)
deg (z)
-- )~. deg (z)-i - v (z).
Hence, for eigenvectors of the Laplacian, the deviation between a node and the centrold of its
neighbors gets larger as the node's degree decreases.
6. A N
OPTIMIZATION
PROCESS
An attractive feature of the degree-normalized eigenvectors is that they can be computed by
an intuitive algorithm, which is directly related to their aesthetic properties. This is unlike the
(nongeneralized) eigenvectors, which are computed using methods that are difficult to interpret
in aesthetic terms. We begin by deriving an algorithm for producing a 1-D layout x E R ~, and
then, we show how to compute more axes.
Our algorithm is based on iteratively placing each node at the weighted centroid of its neighbors
(simultaneously, for all nodes). The aesthetic reasoning is clear. As explained in Section 5, this
principle unifies the Tutte layout [18] and the eigenprojection.
A rather impressive fact is that when initialized with a vector D-orthogonal to 1,~, such an
iterative placement algorithm converges in the dwectzon of a nondegenerate degree-normalized
eigenvector of L. More precisely, this algorithm converges either in the direction of u S or that
of u '~. We can prove this fact by observing that the action of putting each node at the weighted
centroid of its neighbors is equivalent to multiplication by the transition m a t r i x - - D - 1 A . Thus,
the process we have described can be expressed in a compact form as the sequence,
x0 = random vector,
s.t. x~Dl~ -- 0,
x~+l = D-1Ax~.
This process is known as the power-iteration [19]. In general, it computes the "dominant"
eigenvector of D-1A, which is the one associated with the largest-m-magnitude eigenvalue. In
our case, all the eigenvectors are D-orthogonal to the "dominant" elgenvector--l~, and also the
initial vector, x0, is D-orthogonal to ln. Thus, the series converges in the direction of the next
dominant eigenvector, which is either u 2, which has the largest positive eigenvalue, or u n, which
possibly has the largest negative eigenvalue. (We assume that x0 is not D-orthogonal to u 2 or
to u n, which is nearly always true for a randomly-chosen x0.)
In practice, we want to ensure convergence to u S (avoiding convergence to u~). We use the
fact that all the eigenvalues of the transition matrix are in the range [-1, 1]. This can be proved
directly using the Gershgorin bound on eigenvalues [19], since in D - 1 A , all entries on the diagonal
are 0, and the sum of each row is 1. Now it is possible to shift the eigenvalues by adding the value 1
to each of them, so that they are all positive, thus, preventing convergence to an eigenvector with
a large negative eigenvalue. This is done by working on the matrix,
I+D-iA,
instead of the matrix D-1A. In thxs way, the eigenvalues are in the range [0, 2], while elgenvectors
are not changed. In fact, it would be more intuitive to scale the eigenvalues to the range [0, 1],
so, we will actually work with the matrix,
1 ( I + D-1A).
1880
Y KOREN
Function Spectral Drawing G--the input graph, p---dimension
% Thin function computes uS,.. , u p, the top (nondegenerate) elgenvectors of D-1A.
const e ~- 10-8 % tolerance
for k = 2 t o p d o
~k .__ random % random mltlahzat]on
~k
~k
do
%D-Orthogonalize against previous eigenvectors.
forl=ltok-ldo
uk ~ u k (uk) "TDuz l
-
-
end for
% multiply with ½(I + D-1A).
for ~= 1 t o n d o
end for
*- ~
% normahzation
while ~k u k < 1 - e % halt when direction change is negligible
end for
return u 2, ..., uP
Figure 3 The algorithm for computing degree-normalized eigenvectors
If we use our initial "intuitive" notions, this means a more careful process. In each iteration, we
put each node at the average between its old place and the centroid of its neighbors. Thus, each
node absorbs its new location not only from its neighbors, but also from its current location.
The full algorithm for computing a multidimensional drawing is given in Figure 3. To compute
a degree-normalized eigenvector u k, we will use the principles of the power-iteration and the Dorthogonality of the eigenvectors. Briefly, we pick some random x, such that x is D-orthogonal
to ul,... ,u k-l, i.e., xTDu I -- 0,..., xTDu k-1 = 0. Then, if xTDu k ~ 0, it can be proved that
the series,
I (I+D-1A)
2
x,
(I+D-1A
/)
x,
(I+D-1A
x,...
converges in the direction of u k. Note t h a t in theory, all the vectors in this series are D-orthogonal
to u ] , . . . , u k-1. However, to improve numerical stability, our i m p l e m e n t a t i o n imposes the Do r t h o g o n a l i t y to previous eigenvectors in each iteration. T h e power iteration algorithm produces
vectors of diminishing (or exploding) norms. Since we are only interested in convergence in
direction, it is c u s t o m a r y to rescale the vectors after each iteration. Here, we will rescale by
normalizing the vectors to be of length 1.
The convergence rate of this algorithm when c o m p u t i n g u k is d e p e n d e n t on the ratio # k / P k + l .
Running t i m e is significantly improved when replacing the r a n d o m initialization of the vectors
with some well-designed initialization. A way to o b t a i n such a s m a r t initialization is described
in Section 7. A n alternative a p p r o a c h is to embed this algorithm in a multiscale construction.
This is done b y a p p r o x i m a t i n g the original g r a p h using a coarse graph a b o u t half the size, and
then c o m p u t i n g recurslvely the eigenvectors of the coarse graph. These eigenvectors are used to
o b t a i n an initial prediction of the eigenvectors of t h e original graph. Our experience with such a
multiscale s t r a t e g y shows an e x t r e m e l y r a p i d convergence. F u r t h e r details are given in [20].
Drawing Graphs by Eigenvectors
7. S P E C T R A L
DRAWINGS
WITHIN
1881
A SUBSPACE
Most graph drawing methods suffer from lengthy computation times when applied to really large graphs. Hence, a particularly challenging problem is drawing large graphs containing 103-106 nodes, which has gained much interest recently because of the rapid growth of data
collections. (Recent work on accelerating force-directed layout algorithms includes [21-24].) In
general, when using standard eigensolvers the eigenprojection method is very fast compared to
almost all other graph-drawing algorithms. However, calculating the first few eigenvectors of L is
a difficult task that presents a real problem for standard algorithms when n becomes around 10~.
Consequently, we describe an approach that significantly reduces running times, enabling the
drawing of huge graphs in a reasonable time.
Whereas, in eigenprojection, we chose the drawing axes as arbitrary vectors in R n, now, we
would like to constrain the drawing axes to lie within some subspace (vector space) S c_ ~ .
Thus, we require x l , . . . , x p E S. We define a subspace S by some n x m matrix X whose columns
span S (so, S = T~ange (X)). Consequently, we can always denote the axes of the drawing by the
vectors y l . . . , yp E R m , so that
x 1 = X y 1, . . . , X p = X y p.
Clearly, constraining the drawing to lie within a subspace might result in arbitrarily bad
layouts. However, as we are going to show, we can construct subspaces that contain reasonably
nice layouts, so we are not losing much by working only within such subspaces.
7.1. T h e H i g h - D i m e n s i o n a l E m b e d d i n g S u b s p a c e
An appropmate subspace is based on the hxgh-dlmensional embedding (HDE) that we have
already used in [25]. This HDE comprises m axes X 1 , . . . , X m E R ~. In order to construct it,
we choose m p i v o t nodes P l , P 2 , . . . ,pro that are uniformly distributed over the graph and link
each of the rn axes with a unique node. The axis 2dz, which is associated with pivot node p~,
represents the graph from the "viewpoint" of p~. This is done by assigning the 3 th component
of X ~ to the graph-theoretical distance between nodes p~ and j (i.e., the length of the shortest
path connecting the two nodes). Henceforth, we denote this graph-theoretical distance by :Dp~,
so, in symbols,
x ~ (3) = ~p~j.
G(V={1,...,n},E),m
Function HDE
% This function finds an m-dimensional high-dimensional embedding of G
Choose node pl randomly from V
d[1,
, n] ~-- z~
for z = l t o m d o
% Compute the i - t h coordinate using BFS
~p~. *- BFS (G(V, E ) , p , )
for e v e r y 3 E V
db'] ~ mm{db], XZ(9)}
e n d for
% Choose next pivot
P~+I ~-- argmax{oev}{d[J]}
e n d for
r e t u r n X 1,
1~ m
Figure 4 Constructing an m-dimensional HDE
1882
Y KOREN
Function
Orthonormalize
{u 1, . , u m}
% Thin f u n c t i o n o r t h o n o r m a h z e s a set of vectors.
% Also, it o r t h o g o n a h z e s t h e v e c t o r s a g a i n s t I n
c o n s t u ° ~-- ~
II1~I1'
e*-- 0.001
for~=ltomdo
for j = 0 t o z -
1 do
e n d for
i f []u~[[ < e t h e n
a hnearly dependent vector
u z ~---0
else
]lu~ll
e n d if
end for
F i g u r e 5. G r a m - S c h m i d t o r t h o n o r m a h z a t l o n
The resulting algorithm for constructing the high-dimensional embedding is given in Figure 4.
The graph-theoretical distances are computed using breadth-first-search (BFS) [26]. The pivots Pl,P2,... ,Pm are chosen by a heuristic for the k-centers problem, as follows. The first member, Pl, is chosen at random. For 3 = 2 , . . . , m, node P3 is a node that maximizes the shortest
distance from {Pl,P2,... ,P3-1}. This method is mentioned in [27] as a 2-approximation 2 to the
k-center problem, where we want to choose k nodes of V, such that the longest distance from V
to these k centers is minimized. The time complexity of this algorithm is O(m. IEI), since we
perform BFS in each of the m iterations.
We have previously shown that the HDE is a kind of m-dimensional layout of the graph and that
PCA projections of the high-dimensional embedding typically yield race layouts; see [25]. Projections are just a type of linear combination, so restricting the layout to lie inside span ( X 1 , . . . , A'm)
is plausible. In practice, we have found that choosing m ,~ 50 serves very well for producing a
nice layout.
It is important that , ~ 1 , . . . , x m are orthogonal, or at least linearly independent (so they form
a valid basis). Of course, this characteristic can be obtained without altering span ( X 1 , . . . , x m ) .
Moreover, it will be convenient for us that all vectors are orthogonal to 1~, in order to eliminate
the redundant translation degree-of-freedom. We achieve all these requirements by a variant of
the Gram-Schmidt orthonormalization procedure shown in Figure 5. Note that vectors that are
(almost) linearly dependent will get the value 0 and, therefore, should be removed. The entire
orthonormalization process takes O(m2n) time. After completing this process, we take all the
(nonzero) vectors, and arrange them, column-wise, in matrix Af.
7.2. E i g e n p r o j e c t i o n in a S u b s p a c e
As shown in Section 3.1, eigenprojection defines the axes as the minimizers of (14),
P
E
min
x 1, , x p ~
k=l
~5~' (xk)T xk '
subject to'
(xk)T x/ = ~kz,
k=l
= 0,
k,l = 1 , . . . , p ,
k = 1,... ,;
2A 5 - a p p r o x i m a t m n a l g o m t h m delivers a n a p p r o x i m a t e s o l u t i o n g u a r a n t e e d t o b e w i t h i n a c o n s t a n t f a c t o r 5 of t h e
optimal solution
Drawing Graphs by Eigenvectors
1883
In our case, we want to optimize x l , . . . ,x p within the subspace T4ange(Af), and this can be
achieved by replacing them with Afyi,..., Afyp. Hence, (14) becomes,
P
min
yl, ,ypEl~m
E (Yk)T XTLXy k
k=1
p
~ (yk) T XTXy k
k=l
subject to:
(yk)T AfTAfyZ= 5k~,
k,l = 1, ..,p.
Note that here, we do not impose orthogonality of the axes to ln, as it is already achieved
by the fact that 1,~ ~ ~ange (X). We can further simplify the problem by utilizing the fact
that xTAf ----I, obtaining the equivalent problem,
P
E (yk) T XTLAfy k
rain
yi,
k=l p
22
(yk)s yk
k=l
subject to:
(yk)m yl
=
(~kl,
]g, l = 1 , . . . , p .
By Corollary 1, the minimizers of (17), are the lowest eigenvectors of AfTLAf.
To summarize, let us be restricted to a subspace spanned by the columns of the orthogonal
matrix Af. The drawing can be obtained by first computing the p lowest eigenvectors of AfTLAf,
denoted by y i , . . . , yp and then, taking the coordinates to be Afyi,..., Afyp.
Computation of the product XTLAf is done in two steps. First, we compute LX in time
O(mlE[) utilizing the sparsity of L, and then, we compute AfT(LAf) in time O(m2n). Note
that AfTLAf is an m x m matrix, where typically m ~ 50, so the eigenvectors' calculation takes
negligible time (about a millisecond). This is, of course, a significant benefit of optimizing within
a subspace. We recommend that a very accurate calculation be performed. This improves the
layout quality with an insignificant affect on running time. In practice, we invert the order of the
eigenvectors by using matrix,
B=#.I-XTLX,
and compute the highest eigenvectors of B using the power-iteration [19]. The scalar tt is the
highest eigenvalue of AfT LAf that can be computed directly by the power-iteration. Alternatively,
one can set tt to the Gershgorin bound [19], which is a theoretical upper bound for (the absolute
value of) the largest eigenvalue of a matrix. Specifically, for our matrix, this bound is given by
REMARKS.
(i) In a case that we want to use the notion of "degree-normalized" generalized eigenvectors
(i.e., to optimize within the HDE subspace), all we have to do is to D-orthonormalize Af's
columns, instead of orthonormalizing them (this is a slight change to the Gram-Schmidt
process).
(2) When the nodes represent multidimensional points, we can take the original coordinates
as the subspace within which we optimize. One of the benefits is that the resulting layout
is a linear combination of the original multidimensional data. For more details see [28].
1884
Y KOREN
7.3. Results of Subspace-Constrained Optimization
We have validated the performance of optimization within the H D E subspace by experimenting
with m a n y graphs. The results are very encouraging. We obtained quality layouts in a very rapid
time. For example, we provide in Figure 6 the layouts of the four graphs that were previously
shown in Figure 2.
Here, we would like to mention some general comments about the nature of drawings produced
by the eigenprojection. On the one hand, we are assured to be in a global minimum of the energy,
thus, we might expect the global layout of the drawing to faithfully represent the structure of the
graph. On the other hand, there is nothing that prevents nodes from being very close. Hence, the
drawing might show dense local arrangement. These general claims are nicely demonstrated in the
examples drawn in Figure 2. Comparing the layouts in Figure 2 with those in Figure 6, it is clear
(a) The 4970 graph IV] = 4, 970, ]El = 7,400.
(c) The Crack graph [10] IVI = 10, 240, IEl = 30,380
(b) The 4elt graph [10]. IV] = 15, 606, ]E] =
45,878.
(d) A 100 × 100 folded grid wlth central horizontal
edge removed. IV[ = 10, 000, [El = 18,713
Figure 6 Drawings obtained by HDE-constramed eigenprojectlon
Drawing Graphs by Elgenvectors
1885
(a)
(b)
(c)
(d)
Figure 7 Comparison of eigenprojectlon (left) and elgenprojection optimized m
HDE subspace (right) The results are given for the two graphs. Top: the Bfw782a
graph [29] (IVI--782, IE1=3,394). Bottom the Fman512 graph [10] (IYl -- 74,752,
IEI -= 261,120).
that when optimizing within the HDE subspace the layouts are somewhat less globally-optimal
and the symmetries are not perfectly shown. However, an aesthetical advantage of working within
the HDE subspace is that the undesirable locally dense regions are less common, and the nodes
are more evenly distributed over the layout. This can be observed by comparing the boundaries
of the layouts in the two figures or by observing the folded part of the grid in layout (d).
The ability of the HDE-constrained optimization to show delicate details that are often hidden
in eigenprojection layouts stems from the fact that all coordinates in the HDE subspace are integral (up to translation and orthogonalization). We further demonstrate this ability in Figure 7,
where we compared, side-by-side, the results of constrained and unconstrained eigenprojeetion.
Clearly, the new method agrees with the eigenprojection regarding the global structure of the
layout, but provides additional finer details.
1886
Y KOREN
An interesting issue is whether eigenprojection always produces cross-free layouts of planar
graphs. In general, the answer is no. For example, consider the 4elt graph, which is planar as
mentioned in [24]. The eigenprojection layout of this graph, which is displayed in Figure 2b,
contains clear crossings in its top-right portion. An easier example is obtained by connecting two
corners of a squared grid. The resulting graph m planar, but eigenprojection will draw it with
edge crossings. An open question is whether can we define a family of planar graphs for which
eigenprojection produces crossing-free layouts.
t~UNNING TIME. A distract advantage of optimization within the HDE subspace is the sub-
stantial reduction of running time thanks to replacing the n × n eigenequation with an m × rn
eigenequation. We cannot provide a recipe for the value of m, but in all our experiments m = 50
served us well, regardless of the graph's size. Note that unlike all iterative eigensolvers (including
the rapid algebraic-multigrid implementation in [20]), for which the number of iterations depends
on the structure of the matrix, the running time of our algorithm depends only on the graph's
size and is
O(m2n+mlSl).
Moreover, the dimensionality of the drawing has virtually no effect on the running time, whereas
for (unconstrained) eigenprojeetion running time is linear in the dimensionality of the layout
(e.g., time for drawing a graph in 3-D will grow by ~50% relative to drawing it in 2-D).
Table I provides the actual running time of the various components of the subspace-constrained
algorithm, as measured on a single-processor Pentium IV 2 GHz PC. In addition to the total
running time, we also provide the time needed for computing and orthogonalizing the HDE
subspace (in the HDE-titled column), and the time needed for calculating the matrix XTLX (in
the last column).
Table 1 Running time (m seconds) of the various components of the algorithm.
graph
IV[
tel
running time (sec.)
xTLx
total
HDE
516 [10]
516
729
0.02
0.00
0.00
Bfw782a [29]
782
3,394
0.06
0.02
0.00
Fldap006 [29]
1651
23,914
0 06
0.03
0 02
4970 [10]
4970
7400
0.77
0.09
0.64
3elt [10]
4720
13,722
0.77
0.09
0 64
Crack [10]
10,240
30,380
1.80
0.25
1.45
4elt2 [10]
11,143
32,818
1.84
0.28
1.52
4elt [10]
15,606
45,878
2.59
0.44
2.13
Sphere [10]
16,386
49,152
2.91
0.55
2 33
Fldap011 [29]
16,614
537,374
3.28
0.73
2.52
Fman512 [10]
74,752
261,120
8.17
2.83
5.30
Smrpinski (depth 10)
88,575
177,147
13.89
3.19
10 56
grid 317 x 317
100,489
200,344
7.59
3.28
4.24
Ocean [10]
143,437
400,593
25.73
8 00
17 50
8. D I S C U S S I O N
In this paper, we have presented a spectral approach for graph drawing, and justified it by
studying three different viewpoints for the problem. The first viewpoint is based on solving
a constrained energy minimization problem. We shaped the problem so that it shares much
resemblance with force directed graph drawing algorithms (for a survey refer to [2,3]). Compared
with other force-directed methods, the spectral approach has two major advantages.
(1) Its global optimum can be computed efficiently.
Drawing Graphs by Elgenvectors
(2) The energy function contains only O(]EI) terms, unlike the
almost all the other force-directed methods.
1887
O(n2) terms
appearing in
A second viewpoint shows that spectral methods place each node at the centroid of its neighbors
with some well defined deviation. This new interpretation provides an accurate description of
the aesthetic properties of spectral drawing and also explains the relation between "nice" layouts
and eigenvectors in a very direct manner.
We have also introduced a third viewpoint, showing that a kind of spectral drawing is the
limit of an iterative process, in which each node is placed at the centroid of its neighbors. This
viewpoint does not only sharpen the nature of spectral drawing, but also provides us with an
aesthetically-motivated algorithm This is unlike other algorithms for computing eigenvectors,
which are rather complicated and far from having an aesthetic interpretation.
In addition to the theoretical analysis of spectral layouts, we suggested a novel practical algorithm that significantly accelerates the layout computation based on the notion of optimizing
within a subspace. To this end, we described how to construct an appropriate subspace with a
relatively low dimensionality that captures the "nice" layouts of the graph. This way, each axis
of the drawing is a linear combination of a few basis vectors, instead of being an arbitrary vector
in R n (n is the number of nodes). The resulting layout might be the final result of serve as a
smart initialization for an iterative eigensolver.
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